Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls
| dc.creator | Dilworth, S. J. | |
| dc.creator | Howard, Ralph | |
| dc.creator | Roberts, James W. | |
| dc.date | 1998-07-20 | |
| dc.date.accessioned | 2026-07-07T05:25:27Z | |
| dc.date.available | 2026-07-07T05:25:27Z | |
| dc.description | A real valued function $f$ defined on a convex $K$ is anemconvex function iff it satisfies $$ f((x+y)/2) \le (f(x)+f(y))/2 + 1. $$ A thorough study of approximately convex functions is made. The principal results are a sharp universal upper bound for lower semi-continuous approximately convex functions that vanish on the vertices of a simplex and an explicit description of the unique largest bounded approximately convex function~$E$ vanishing on the vertices of a simplex. A set $A$ in a normed space is an approximately convex set iff for all $a,b\in A$ the distance of the midpoint $(a+b)/2$ to $A$ is $\le 1$. The bounds on approximately convex functions are used to show that in $\R^n$ with the Euclidean norm, for any approximately convex set $A$, any point $z$ of the convex hull of $A$ is at a distance of at most $[\log_2(n-1)]+1+(n-1)/2^{[\log_2(n-1)]}$ from $A$. Examples are given to show this is the sharp bound. Bounds for general norms on $R^n$ are also given. | |
| dc.description | 39 pages. See also http://www.math.sc.edu/~howard/ | |
| dc.identifier | https://arxiv.org/abs/math/9807107 | |
| dc.identifier | http://arxiv.org/abs/math/9807107 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/77182 | |
| dc.subject | Metric Geometry | |
| dc.subject | 26B25 52A27 (primary), 39B72 41A44 51M16 52A21 52A40 (secondary) | |
| dc.title | Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls | |
| dc.type | text |